Was Pythagoras the First to Discover Pythagorasís Theorem?May 1, 2013
Figure 1. An Egyptian surveyor set out with ropes and rods to measure land area.
Figure 1. An Egyptian surveyor set out with ropes and rods to measure land area.
According to Brown University mathematician David Mumford, the answer to the question is an emphatic "No!" On February 27, 2013, in a public lecture at the Institute for Mathematics and its Applications at the University of Minnesota, Mumford showed how ancient cultures, including the Babylonians, Vedic Indians, and Chinese, all proved the beloved formula long before the Greeks. He argued that the theorem is ultimately the rule for measuring distances on the basis of perpendicular coordinates. This comes up naturally in calculations of land area for purposes like taxation and inheritance, as shown in Figure 1. He further suggested that the Greeks' love of formal proof may have contributed to the Western belief that they discovered what Mumford calls the "first nontrivial mathematical fact."
Along with Pythagoras's theorem, Mumford discussed the discovery and use of algebra and calculus in ancient cultures. One of his key points is that deep mathematics was developed for different reasons in different cultures. Whereas in Babylonia algebraic "word" problems were posed seemingly just for fun, the Nine Chapters on Computational Methods, considered the Chinese equivalent of Euclid's Elements, was compiled in about 180 BCE for very practical applications--among them Gauss-ian elimination for solving systems of lin-ear equations, which the Chinese carried out using only counting rods on a board (Figure 2). Riemann sums grew naturally out of the necessity for estimating volume. Mumford suggested that Vedic Indians even pondered problems of limit in integral calculus.
Figure 2. Gaussian elimination, performed in China during the Han dynasty, with only rods and a board.
Contrary to Western historical belief, Mumford showed, the West did not always lead in mathematical discovery. Apparently, the origins of calculus sprang up totally independently in Greece, India, and China. Original concepts included area and volume, trigonometry, and astronomy. Mumford considers the year 1650 a turning point, after which mathematical activity shifted to the West.
Mumford's presentation runs counter to current texts on the history of mathematics, which often neglect discoveries occurring outside the West. He showed that purposes for which mathematics is pursued can be very culturally dependent. Nevertheless, his talk points to the fundamental fact that the mathematical experience has no inherent cultural boundaries.
Mumford, a professor emeritus in the Division of Applied Mathematics at Brown University, has worked predominantly in the area of algebraic geometry and is a leading researcher in pattern theory. Mumford received a Fields Medal in 1974; his more recent awards include the Shaw Prize (2006), the Steele Prize for Mathematical Exposition (2007), the Wolf Prize (2008), and the National Medal of Science (2010).
Anna Barry is a postdoctoral fellow at the Institute for Mathematics and its Applications at the University of Minnesota. Following up on her coverage of David Mumford's IMA lecture for SIAM News, she conducted the interview transcribed below.
An Interview with David Mumford
You have done highly distinguished work in both pure and applied mathematics, from algebraic geometry to computer vision, and now the history of mathematics. How did your career path evolve?
I've always been interested in things besides math. Growing up, I was very interested in physics and worked for Westinghouse on the atomic submarine reactor while a student. Around 1956, listening to the lectures of George Mackey, Lars Ahlfors, and Oscar Zariski, I fell in love with abstraction, with the idea of a secret garden containing amazing things that could only be seen with the mind's eye but never touched: pure math. However, I always had an interest in other questions. I proved a theorem in algebraic geometry in the 1980s that was the culmination of a long series of efforts, and I thought, If I'm ever going to pursue these other things that I had wanted to as a student, then now would be the time.
During the 1950s, I had read everything I could about the brain and how it operates. It felt as though there was a real possibility of taking a limited aspect of the capabilities of the brain, specifically of understanding two-dimensional images and making a model of a three-dimensional world with them.
As for history, there was a great group of mathematicians in India that I visited in Bombay quite a lot. In the 1990s I talked to some people about the history of Indian math, read about it, and one thing led to another. The lecture I gave last night was a culmination of trying to put together the side of history that I find most fascinating---that similar things were discovered in many cultures but always in a different context. The culture made a very strong mark on the mathematics.
Do you believe that cultural differences affect the way that mathematicians communicate even today?
No, now we all speak the same language on a scientific level. I think it's fantastic that these barriers have been removed. The whole intellectual enterprise is so vast these days.
What are some of the difficulties you faced as you moved from one area of mathematics to another?
There was a period of about ten years before I felt like I had my feet on the ground in computer vision. Trying to find the right mathematical tools was difficult, but I gradually got a sense for what these might be. I've always thought that my skills were those of a mathematician, but I loved the idea that they could be used to model something like a cognitive skill.
What advice can you share with other mathematicians attempting to break into new areas of mathematics?
I think it's a matter of personality and a question of comfort zone in many ways. I think I've had an adventurous streak. Perhaps I'm less inhibited. I acknowledge that I didn't make the same impact on vision that I did in algebraic geometry. Vision is a much bigger field, with people pursuing it from all sorts of different directions. I felt my contribution was to raise questions the way mathematicians would, and this point of view was useful. I try to understand on a deeper level as much as possible what the fundamental problems are.
The IMA also works to bring applications to the attention of pure mathematicians. At Brown, we have started a similar institute, ICERM; one of the narratives that started ICERM was that arithmetic algebraic geometry led to ideas about encryption and the people involved in this formed a kernel and it blossomed. In the end it's a question of what your skills are and what you absolutely love doing. People are different. I think what you should do work on is what you look forward to doing and what you could spend a whole day working on by yourself. It must be the thing that you're really good at and what you really want to do. You have to find your comfort zone, and it's different for each person.
The fact that Pythagoras's theorem was discovered by several different cultures begs the following question: Do you believe that mathematical truths are in existence and to be discovered, or are they constructs of human imagination?
I think if you're a mathematician and encounter any problem, you can't really do work on it unless you think somehow that the answer is out there---that it's been there all along, and you're trying to find a way to unlock some door and beyond that door you'll see the answer. Psychologically, in order to come to grips with a problem, it must become so real to you that you can't get it off your mind. And if you're lucky, you stumble on what the Indians called the "yukti," the yoke that connects the ideas which solve the problem.
Is there anything else you'd like to add?
Yes. Having worked in both pure and applied mathematics, I have been very conscious of the barriers between these two sides. Actually, I don't like to think of them as two sides. I like to think of pure math as being a core, and applied math as being a whole series of subjects arrayed around the core that bring these mathematical tools into all sorts of applications.
To lower the barriers, pure and applied mathematicians both have important jobs. Pure mathematicians should include more explanation and motivation in their papers and talks, and discuss the simple cases of their results. Simplify things---there is nothing wrong with that. The applied mathematician has the difficult job of looking at a problem in context with no explicit mathematics and trying to see what kind of mathematical ideas are under the surface that could clarify the situation. I think the most successful applied mathematicians are those who look in both directions, at the science and at the math.
You can't become too attached to one way of looking at things. Applied math has always rejuvenated pure, and theorems in pure math can unexpectedly lead to new tools with vast applications.---AB